Multivariate Normal Approximation for Functionals of Random Polytopes

Consider the random polytope that is given by the convex hull of a Poisson point process on a smooth convex body in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document}. We prove central limit theorems for continuous motion invariant valuations including the Wills functional and the intrinsic volumes of this random polytope. Additionally we derive a central limit theorem for the oracle estimator that is an unbiased and minimal variance estimator for the volume of a convex set. Finally we obtain a multivariate limit theorem for the intrinsic volumes and the components of the f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {f}$$\end{document}-vector of the random polytope.